To find dz/dx and dz/dy using implicit differentiation for the equation e^(6z) = xyz, we first need to differentiate both sides of this equation with respect to the chosen variable while treating the other variables and z as functions of that variable.
Let’s start by differentiating the left side:
- The left side is e^(6z). When we differentiate, we’ll apply the chain rule:
- d/dx(e^(6z)) = e^(6z) * (d/dx(6z)) = e^(6z) * 6(dz/dx)
Now for the right side, xyz:
- For this, we’ll use the product rule. The product rule states that d(uvw) = uv’ + vu’ + uvw’:
- d/dx(xyz) = y*z + x*(dz/dx)*y + x*y*(dz/dx)
Putting it all together, we set the derivatives equal to each other:
e^(6z) * 6(dz/dx) = y*z + x*(dz/dx)*y + x*y*(dz/dx)
Next, we collect all the dz/dx terms on one side:
e^(6z) * 6(dz/dx) – x*y*(dz/dx) = y*z
Now, factor out dz/dx:
dz/dx * (e^(6z) * 6 – x*y) = y*z
Finally, solve for dz/dx:
dz/dx = y*z / (e^(6z) * 6 – x*y)
Now, to find dz/dy, we’ll perform a similar process, but differentiate with respect to y instead:
- Differentiate e^(6z):
- d/dy(e^(6z)) = e^(6z) * 6(dz/dy)
- Differentiate xyz:
- d/dy(xyz) = x*z + x*y*(dz/dy)
Setting these equal gives:
e^(6z) * 6(dz/dy) = x*z + x*y*(dz/dy)
Next, we arrange it similar to before:
e^(6z) * 6(dz/dy) – x*y*(dz/dy) = x*z
Factor out dz/dy:
dz/dy * (e^(6z) * 6 – x*z) = x*z
Finally, solving for dz/dy gives:
dz/dy = x*z / (e^(6z) * 6 – x*y)
In summary:
- dz/dx = y*z / (e^(6z) * 6 – x*y)
- dz/dy = x*z / (e^(6z) * 6 – x*y)